Sound Production in Pianos

By A.H. BENADE

In the second half
of chapter 16 we examined the interplay between the physics of vibrating
strings and the tuning behavior of various kinds of keyboard musical
instruments. In the present chapter we will turn our attention to the way in
which the strings of a piano, harpsichord, or clavichord communicate their
carefully tuned vibrations to the soundboard and thence to our ears. Thus we
will be focusing our attention on the nature of the sounds produced by these
instruments. In broad outline, the chapter will begin by examining the
requirements that must be met by a string belonging to a note in the middle of
the piano keyboard. This is followed by an account of what is accomplished by
the use of more than one string per note and a description of the changes and
compromises necessary for satisfactory production of lower and higher tones.
Having dealt with sounds produced on the piano, we will be ready in chapter 18
to adapt our understanding of these basic principles to the clavichord and the
harpsichord.

17. 1. The Soundboard As Seen by the Strings; The
Concept of Wave Impedance

Anyone looking into a grand piano will notice that
each note in its scale has one or more strings stretched across a sound­board
in the manner shown in figure 17.1. The so-called vibrating length of the
string extends from a rigid capo dasrtro
bar (or from a fixed agraffe) which
is found at the keyboard end, to the bridge, which is elastically supported by
a broad, thin soundboard. A hammer is arranged to excite this length of string
by striking it at a suitably chosen point near the fixed end. Not shown in the
figure is a felt damper that normally rests on the string near the hammer to
keep it from vibrating when it is not in use.

In chapters 7 and 8 we learned
some of the basic principles guiding the excitation of string modes by hammers
of various sorts acting at various distances from the fixed end. Now it is time
to look at the way in which these characteristic vibrations of the string are
communicated to the soundboard and thence to the concert hall. The string and
the soundboard meet by way of the bridge, so we need to know what sort of a termination
the string "sees" as it looks at the place where it runs over the
bridge. The bridge actually functions acoustically as a part of the soundboard,
which is attached to its lower side. The soundboard is a two ­dimensional
wave-carrying medium of the sort we first met in chapter 9. Carefully profiled
ribs run across the grain on the underside of the soundboard to make its
stiffness approximately the same across the grain as it is lengthwise. We have
already acquired at least a general idea of the way in which such a uniform two­
dimensional object will respond to an excitation applied at some point on its
surface (see figs. 10.13 and 10.14). In section 16.5 we also learned a little
about how the driven motion of the soundboard can react back on the string to
alter its natural frequencies.

There is more to what the string
sees at the bridge end than just bridge and soundboard. There are some 240
other strings running over the top surface of the bridge, and these also form a
kind of two­

dimensional wave-carrying medium that is
"visible" to our vibrating strings, albeit in a more limited way
because waves are not able to run easily in a crosswise direction from string
to string. (There is only the bridge to connect them to each other, with no
ribs to equalize what we might in this case call the cross-grain and
along-the-grain properties of the system.) The fact that most of the strings
are damped by pieces of felt need not concern us at present, any more than does
the question whether the edges of the Sound­board function as hinged or as
clamped boundaries.

The playing string, the bridge-
plus­- soundboard, and the sheet of silent but down bearing strings can each be
thought of as a wave-carrying medium. Acoustical theory tells us that any
wave-carrying medium can be characterized fully by two specifications: the
velocity with which waves are propagated along the medium, and the wave
impedance. We will briefly review the first and more familiar of these before
considering the idea of wave impedance.

When any sort of acoustic disturbance is made at one
point in a wave-carrying medium, it takes a little while for the disturbance to
make its appearance at another point a little distance away. The rate at which
the disturbance travels from its source to the point of observation is what is
known as the speed of sound or wave
velocity (for example, we learned in sec. 11.8 that the speed of sound in
air is about 345 meters/sec). The wave velocity always depends on the springiness
or elasticity with which one small part of the medium acts on its neighbours
during a disturbance; the wave velocity depends also on the inertia of the
material (i.e., the amount of mass belonging to each of these small parts).

These are related to each other by the following
formula:

wave velocity =Ö springiness / inertia

Question 2 in the final section of this chapter will
help you understand why this formula for the wave velocity (the speed of sound)
looks so remarkably like the formula found in section 6.1 for the natural frequency of oscillation of a spring-and-mass
system.

The second concept we need for an
understanding of how a struck string com­municates with the soundboard is the
idea of wale impedance. When a
disturbance is set up in some medium and travels to the boundary between it and
some other medium (as when disturbances travel along a slender wire to a
thicker wire or a soundboard), a certain fraction of the disturbance is
transmitted into the new medium and the remainder is reflected back into the
original medium. The amplitudes of the reflected and the transmitted waves, and
also the amounts of energy carried by them, all depend on the ratio of the wave
impedances of the two media.' If these impedances are very different, there is
almost complete reflection, with only a small share of the total energy being
sent on. On the other hand, if the two media have wave impedances that are
approximately equal, then there is very little reflection and the disturbance
is almost completely transmitted across the junction. It turns out that wave
impedance depends on the same two properties of the medium as does wave
velocity, although they are arranged differently, thus:

In order to keep things clear for the lest
technically oriented readers of this book, I hate chosen to use the slightly
oldfashioned name wave impedance rather than the more current terns characteristic
impedance. Everywhere else in this book,
the perfectly customary adjective characteristic has been resend for use as a way to advertise certain attributes of
some mode of vibration belonging to a particular finite system of springs and masses. Thus, for a
given system, its entire behavior can be understood in terms of its modes of
vibration, each of these having its own characteristic frequency of vibration,
its own characteristic vibrational shape, and its own characteristic
(internally caused) damping. There characteristic properties are determined
jointly by the nature of the vibrating medium and by the way in which its
boundaries are constrained. It is only under very special circumstances that one finds a characteristic vibration taking
place in an infinitely extended system, and then it exists only in a restricted
region of it. The watt impedance, on the other hand, is a way of specifying
(along with the wave velocity) one of the attributes of the wavecarrying medium
itself, without reference to its boundaries. As a matter of fact, one of the
easy ways to measure a wale impedance is to experiment on a very extended piece
of material and to conclude the measurements before any echoes can be returned
by its boundaries (see sec. 17.4 below for an example of this). I might remark
that those of us who use today's more conventional terminology in our daily
work are in the habit of identifying the special attributes of bounded systems
by use of the German prefix eigen­ in
place of the English word characteristic that we employ in this book.

Let us
illustrate the ideas of wave velocity and of wave impedance by considering the
case of waves on a flexible string made of some material whose density is d and
whose radius is r (i.e., crosssectional area = šr2). The string is long, and it is kept under a tension T.

wave velocity = Ö T/ šr2d = (1/r) Ö T/šd

wave impedance = Ö (šr2d) T = r ÖšTd

Here the
tension T serves to supply the springiness, and the product šr2d will be recognized as the mass per unit length, which is a measure
of the relevant inertia property of the string. Notice that we can trade
tension for density or radius while keeping the impedance the same, but it is
not possible at the same time to preserve the speed unchanged.

An analogous but
somewhat simplified formula for the wave impedance of a soundboard at its
driving point is:

wave impedance

2

= tÖ Yw dw x a , (a= numerical constant)

Here t is the
thickness of the board, dw is the density, and Yw is the modulus of
elasticity for the wood [2]. We will assume that the ribs and bridge have been
so designed that they properly take care of the difference in stiffness in the
two directions relative to the grain, and chat the thickness t is also properly
averaged to take these extra pieces into account. I should remark that the wave
impedance of the board taken by itself is very considerably larger than that of
a string. Because it will have little utility for us here, we will say nothing
about the wave velocity in the soundboard beyond remarking that it is
proportional to and that it has the unusual feature of having a different value
at different frequencies.

We must also consider, besides the
playing string and the soundboard, the aggregate influence of the damped and
inactive strings. Their influence is best though[ of in two parts. The simplest
but least important part is the wave impedance of the collection considered as
a peculiar two-dimensional sheet; this turns out to depend on the strings'
spacing along the bridge, and it has a magnitude only three or four times the
impedance of a single string. The second and rather larger influence comes from
the way the downward pull of the slanting
strings between bridge and hitch pins alters the elasticity of the otherwise
slightly arched soundboard, subtly modifying the soundboard wave impedance
formula given above.

The string layout between the
bridge and the hitch pin is illustrated in the top part of figure 17.2. This
silent portion of the string (which is provided with a damping strip of felt)
has a length Q and a "downbearing" P that is carefully pro­portioned
to vary along the scale of any properly made instrument. Makers of the finest
instruments find that the down­bearing must be meticulously adjusted string by
string on each individual piano, as a part of its final regulation. Errors in
the trend of relationship among P, Q, and the string tension can cause as much
trouble to the overall sound of the piano as can errors in the stiffness and
curve of the bridge, or in the thickness of the soundboard. If the ratio P/Q is
locally too small, the instrument acts somewhat the way it would with a thin
spot in the soundboard. Notice that the downbearing is not simply a matter of
getting adequate contact between the bridge and the strings; the string tension
acting together with the offset on the bridge where the string runs zigzag past
two steel pins is already quite sufficient for this contact, as is suggested by
the lower part of figure 17.2.

The wave impedance
ratio between the struck string and the soundboard must be chosen to meet two
conflicting requirements. First of all, there must be sufficient transmission
of vibratory energy from the string to the soundboard that our ears are
ultimately provided with a sound of satisfactory loudness. If the soundboard
were a plate of steel 4 cm thick instead of a wooden board about 1 cm thick,
its wave impedance would be increased several hundredfold and we would hear
almost nothing from the soundboard, nor would the string produce much sound
directly in the air. If on the other hand the disturbance excited on the string
by the hammer were communicated to the soundboard at too rapid a rate, these
vibrations would die down so quickly that we would hear little more than a
tuned thud, a louder version of what is produced by hitting a note while a
wadded handkerchief is firmly pressed against the vibrating part of the string
next to the bridge. We also want the soundboard impedance to be high enough
that its resonances will not play an unacceptably large role in the tuning of
individual string modes, a phenomenon chat we met in section 16.5.

Digression on the Vibrational Modes of Segments of a
Larger System.

In chapter 6 and in the latter half of
chapter 10 the idea was developed that any finite-sized system of springs and
mattes would hair its own Particular set of characteristic vibrational modes,
these modes being attributes of the system as a whole. This implies that it
makes no sense to consider apart from the whole a particular matt or even a
subset of mattes at a separate vibrational system. In this chapter I have
apparently violated this principle of the unified behavior of a complete system
by discussing the string modes and the soundboard modes at though these were in
fact separable. Let us tee why this very convenient separation of ideas proves to be acceptably accurate at a piece of
physics. If a certain complete system (e.g., string and soundboard) consists of
two parts or regions with drastically different wave impedances, the
communication of vibrations from one of these parts to the other it small
enough that the two behave very much at though they were fully isolated. When
this condition it met, then, it is possible to pick out of the complete set of
characteristic erodes a subset in which the overall vibrational shapes ordain
that the predominant share of the vibration takes place in the high-impedance
region, while the remaining modes involve chiefly the rest of the system, which
it constructed of low-impedance material. Once the approximate vibrational
shapes associated with each region alone are well understood, it is then easy
enough to correct for the mutual influence of the two regions. It is in
precisely this spirit that we corrected the string erode frequencies for the
effect of soundboard resonances in section 16.5; the sound­board has a wave
impedance to much higher than that of the strings that we are justified in
thinking of them at quasi-separate entities. Notice, however, that the wave
impedances of the bridge and the soundboard are similar enough that we would
not be justified in dealing with them separately-the two act together with the
ribs at a single wooden vibrating system (see sec. 9.5 for another example of a
two- part system that cannot be dealt with piecemeal).

17.2. The Proportions of a Mid-Scale Piano String and
the Necessity for Multiple Stringing

In section 16.5 we learned that the stiffness of real
strings gives rise to a slight inharmonicity in the ratios between their
characteristic frequencies and that this inharmonicity was less in long, taut,
thin strings than in short, slack, thick ones. We have seen how a small amount of string-type
inharmonicity serves a useful purpose-it can help disguise the necessary errors
of keyboard temperament or can even convert some of these errors into musical
virtues. Moreover, numerous ex­periments have shown that a certain amount of
inharmonicity is necessary if the listener is to be satisfied that what he
hears is an impulsively excited string sound. Nevertheless the history of
keyboard instrument development from the earliest times reveals an intense
though not always conscious interest in reducing the inharmonicity. Because of
this, we will begin our discussion of the proportioning of midscale strings by
postulating that their tension is to be made as large as is reasonably
possible, short of breaking the string. On the basis of this choice, the
vibrating length L of the string turns out to be a fixed length that is
independent of the string's thickness. This is the reason that the length of
the C4 string is close to 62.5 cm for all stee ­strung pianos. The minimum
inharmonicity associated with a string tightened nearly to breaking tension
depends in a simplified way on its radius r and length L as follows (compare
with the formula for J given in sec. 16.5):

Jmin= r2 / L2x
a,(a numerical constant)

This suggests that we should use the thinnest possible
string, since L has already been fixed by the frequency requirements laid down
for the string. However, if we make the string too chin we are speared on the
other horn of our dilemma. The transmission of vibration from our string to the
soundboard is proportional to the wave impedance ratio, and so depends on the
wire radius and soundboard thickness (as influenced by the ribs) in accordance
with the expression:

This relationship holds only if the string tension is
always maintained fairly close to breaking. The equation indicates that making
the wire thin will mean that it will be able to drive the soundboard to only a small
fraction of its own amplitude, so that only very weak sounds will be radiated
into the room.

Some numerical values for the
sound­board and strings of a real piano should be of interest at this point. A
good piano has a soundboard made of beautifully finished spruce that has a
density dw close to 0.4 grams/cm3. The soundboard is often tapered and is
generally thinner at the bass side, but in the main its thickness is a little
less than 10 mm. The radius of the C4 string is close to 0.5 mm, and its density
d is close to 7.8 grams/cm3. A single such string on a piano having a
soundboard of this description sustains its tones very acceptably and shows
tuning behavior almost identical with that described in chapter 16. However,
the loudness of the sound of the single string is inadequate and the tone lacks
a certain liveliness that we have become used to in pianos having three strings
instead of one for most of the keyboard notes. An obvious way of simultaneously
meeting the least- inharmonicity requirements (which call for thin strings) and
the loudness requirements is to use several strings, each of which will have
acceptable inharmonicity and each of which can join with the others in driving
the soundboard to a greater vibrational amplitude. The physics of the
multistring piano note turns out to have surprising aspects that lead to two
important features of the tone of a piano; a description of these matters is
the subject of the next section.

17.3. The Effect of Multiple Stringing on the Sound of
the Piano

We will introduce ourselves to some of the
consequences of multiple stringing on a piano with the help of experiments you
can easily try. Repeatedly strike the C4 key of a piano while alternately
pressing and releasing a finger (or pencil eraser) against two of the three
strings, so that part of the time only one string is free to vibrate and the
rest of the rime all three strings are sounding. With any reasonably well-tuned
piano, the perceived loudness at your ears (expressed in sones) should be roughly
40 percent higher when three strings are active than when only one is producing
a sound (see curve B of fig. 13.5), which is a quite significant change. The
next experiment con­sists in verifying in a crude and informal way that the
total audibility time of the decaying tone is roughly the same whether three
strings are active or only one. So far everything appears to be in accordance
with our expectations. We also notice that the tone is a little thinner and
perhaps less interesting when only one string is allowed to sound than it is
when all three are set into vibration. To be sure, if the piano is badly out of
tune the three strings will beat against one another to give the jangling sound
conventionally associated with a barroom piano, while on a freshly tuned
instrument there is only a hint of beats among the lower partials and a
pleasantly shimmering suggestion of beating among the higher ones.

In 1959 Roger Kirk of the
Baldwin Piano Company reported the preferences of a large group of people for
the tuning relationship among the three strings of each so-called unison of a
piano.' He found that :

the most preferred
tuning conditions . . . are 1 and 2 cents maximum deviation among the strings
of each note in the scale. Musically trained subjects prefer less deviation . .
. than do untrained subjects. Close agreement was found between the subjects'
tuning pref­erences and the way artist tuners actually tune piano strings.

He also found that a piano tuned so that the group of
strings for each note of the scale covered a spread of 8 cents was acceptable
to many listeners, and that the overall spread between the lowest and highest
frequency strings was of more importance than the tuning of the intermediate
string. The beat frequencies between the first five components (partials) of
two C., strings tuned 2 cents and 8 cents apart are: )s 335 Hz beat that one
uses in setting the equal-temperament fifth to G., (see sec. 16.6, part D).
Note that partial 2 of the G., strings will have a similar bearing rate to obscure
further the departure from just tuning. With the 8-cent interstring spread, on
the other hand, the fifths be­come pretty diffuse.

Let us turn now to the interval of
a major third in equal temperament. Using a 2-cent detuning, the fifth
component group of Ca has within it a 1.5-Hz maximum beating frequency, as does
the fourth component group of the note E., if its strings similarly have a
2-cent detuning spread. Taking these together we see the possibility of beat
frequencies as high as 1.5 + 1.5 = 3 Hz among the components upon which the
interval is chiefly based. In section 16.7, we learned that the beating rate
for a piano tuner's third in equal temperament is about 8 Hz, a little more
than twice the smearing produced by the detuned unison. If the spread among
members of a three-string "unison" were increased to 8 cents, the
beating would become rapid enough to

Component:12345

2-cent difference:0.300.610.911.201.50 Hz

8-cent difference:1.212.423.534.806.10 Hz

Notice first of all that with the
2-cent detuning the beating rate for the first pair of partials is quite slow,
as are those for the second and third pair of partials. As a result the tone
sounds reasonably smooth when played by itself. The 8-cent spread gives a
rather brighter sound, but it is not yet the sort of jangle one gets with a
spread of 15 to 20 cents.

When we use a 2-cent detuning between strings, the
0.91-Hz beat frequency belonging to its set of 3rd components is just able to
cover up the 0.89­ Hz beat that one uses in setting the equal-temperament
fifth to G4 (see sec 16.6, part D). Note that partial 2 of the G4 strings will
have a similar bearing rata to obscure further the departure from just tuning.
With the 8-cent inter string spread, on the other hand, the fifths be come
pretty diffuse.

Let us turn now to the interval of
a major third in equal temperament. Using a 2-cent detuning, the fifth
component group of C4 has within it a 1.5-Hz maximum bearing frequency, as does
the fourth component group of the note E4 if its strings similarly have a
2-cent detuning spread. Taking these together we se, the possibility of beat
frequencies as high as 1.5 + 1.5 = 3 Hz among the components upon which the
interval is chiefly based. In section 16.7, we learned that the beating rate
for a piano tuner's third in equal temperament is about 8 Hz, little more than
twice the smearing produced by the detuned unison. If the spread among members
of a three-string "unison" were increased to 8 cents, the bearing
would become rapid enough to drown the temperament error completely. Clearly
there is a trade-off of musical virtues between the two kinds of unison spread
as one compares various musical intervals. In any event we have provided
ourselves with another reason stringed keyboard instruments are so well-adapted
to musical performance, despite the problems with fixed pitch that at first
seemed insurmountable.

As a practical matter it proves to
be exceedingly difficult to tune a set of unison strings to a true zero-beat
condition (one even meets cases where it is literally impossible to do so). The
question arises then whether or not people's preference for a slight detuning
of the unisons is simply a favorable response to the most familiar type of
sound, or whether something more fundamental is involved. Kirk finds that piano
tuners and musicians are unanimous in their verdict that too-close tuning gives
a tone that not only sounds dead but dies away too rapidly. Laboratory
measurement confirms the auditory impression we gained in our initial
experiments that slightly detuned (normal) strings die away in about the same
total length of time as a single one of these strings when the other ones are
prevented from vibrating. However, when three strings are tuned exactly together they will actually die
away much more rapidly. The presence of other precisely in­tune strings
encourages each string to transfer its vibration more rapidly to the soundboard
and thence to the room! Let us first make use of our knowledge of wave
impedance to verify its consistency with these observations and then go on to
an example of the same kind of physics displayed in an everyday experience far
removed from acoustics.

In section 17.1 we learned that
the wave impedance of a string is equal to the square root of the product of
tension T and mass per unit length (šr2d). How do we find the
corresponding impedance for a triplet of identical strings acting together? The
top part of figure 17.3 indicates the appearance of our three strings as they
are normally seen in a piano. The middle part of the diagram shows them moved
so close together that they are on the verge of touching. If they were identical- tuned strings, they would
stay precisely in step with one another, and there would be no frictional or
other force acting between them to change things in case they did touch. In
other words, the three closely spaced strings will behave exactly like their
more separate cousins. In particular, the aggregate impedances are the same in
both cases. The bottom part of figure 17.3 shows the last step in our imaginary
set of transformations: here the strings are fused together into a ribbon like
whole, with no change of total mass or tension. An extension of our former
reasoning shows that this new sort of string also retains the acoustical
properties of its ancestor at the top-as long as we confine ourselves to
vibrations of the normal type (up and down, as shown in the diagram).

Having done a little thinking
about three strings acting precisely together, we are now ready to calculate.
Clearly, the total tension acting on our composite string is three times the
tension acting on each of the original strings, so we must write 3T under the
square root sign where formerly there was a T. Similarly, any short length of
the composite has precisely three times the mass of a corresponding length of
ordinary wire, so we must also write 3(šr2d) in place of šr2d in the formula. Putting all this together, we get:

(wave impedance
of a tricord )= 3 x (šr2d) x 3T=

= 3 x ( wave
impedance of a single wire )

This shows us that three strings acting precisely
together produce a threefold increase in the wave impedance, and thus a
threefold increase in the amplitude of the bridge motion, which ultimately
leads to a threefold reduction in the decay time of the vibration. You might
find it worthwhile to deduce this last assertion on the basis of the principles
outlined in section 6.1.

The expected difference in sound
between a struck single string and a perfectly tuned triplet of strings is not
hard to figure out on the basis of what we have just learned. First of all, the
tone of the precisely tuned triple strings will die away much more quickly,
which matches actual experience. Second, we would expect on the basis of curve
A in figure 13.5 that the perceived loudness of the fundamentals of thetone (as expressed in sones) would be very
nearly doubled because of the threefold increase in source (soundboard)
amplitude. The cone would not actually appear this much louder, however,
because a short or decaying sound always sounds less loud that a steady one. In
the three­ string case the increased rapidity of the decay partially offsets
the perceived effect of the larger amplitude.

We seem by now to have left the
slightly detuned strings of a real piano in a sort of unexplained limbo between
the single string and a perfectly tuned triplet. The true behavior of detuned
triplets will be easy to understand once we have looked ac the everyday example
I promised a few paragraphs ago. Suppose you have undertaken to push your
friend's small car along a fairly level road. If the rolling friction of the
car is large, you may find is barely possible to keep the vehicle rolling, and
yet you will be able to move the car quire a distance under these conditions
without much strain and without becoming winded. Suppose on the other hand that
you have acquired a helper in the pushing, so that the two of you together can
get the speed up to a fast walk. Pushing at this faster pace will soon leave
you winded and panting for breath, even if you are not pushing any harder as an
individual than you were during the solo performance. The point is this: the
energy you expend in pushing with a certain force over a given distance will be
spent in a much shorter rime if your friend helps you make the trip more
quickly. The rare at which you work
is increased because of the cooperative presence of your friend.

The translation of this example to
the case of vibrating strings is easy: one string pulling up and down on the soundboard
and moving it corresponds to you pushing on the car alone. If two strings are
less than precisely in tune with one another, the situation is like the case
where your car-pushing friend sometimes pushes with you and sometimes pushes in
opposition to you. In a semi-disorganized situation like this there is no
absolute coherence to the undertaking and the aggregate accomplishment is
simply equal to the sum of the separate contributions.

Daniel Martin and his research
group at Baldwin Piano Company have shown that a very characteristic feature of
the sound from a piano is a dual decay pattern. This is the second musically
important result of the use of multiple strings. A blow from the hammer starts
all three strings off exactly in step with one another, so that they radiate
strongly to the outside world. Initially, then, each partial dies away quickly
at about the rate expected for strings that are in precise unison. However,
because of their slight detuning from one another, they soon get out of step,
so that we might say that there are eventually three solo performances. The
vibration of each string then decays on its own in isolation at the
single-string race, and close cousins to ordinary beats are produced for us to
hear.

When the strings of the C4 note on
a good piano are tuned to a total spread of about 2 cents, the net sound
pressure due to all the partials (see sec. 13.2) shows the presence of fast
decay for about 1 second our of the coral time of 20 seconds (crudely speaking)
that is required for the net pressure amplitude to be reduced to 1/ 1000th of
its initial value.

I will close this section with a
brief explanation of the pitch rise that is often perceived in a piano tone as
it dies away. To begin with there is a clearly audible change in tone quality,
explainable in part by the fact that the lower-frequency partials become
unimportant and then inaudible more quickly than do the higher partials, simply
because of the greater sensitivity of the ear at high frequencies (see fig.
13.3). Furthermore, the amplitude of the lowest partial generally falls away
more quickly than the higher partials, chiefly because the slow beating rare
between the strings for this component keeps their vibrations in step for a
longer time, during which they suffer the accelerated decay characteristic of
the cooperative effect. This gives us an additional, mechanical reason to
expect a listener's attention to transfer itself to the higher partials of a
decaying tone. Because of string inharmonicity, these higher partials heard by
themselves imply a higher pitch than that which our ears assign when they base
their "calculation" on the lower partials (see also sec. 16.7).
However, the decay patterns of individual notes of a keyboard scale differ
enough from note to note, even on a very fine instrument, that we should not
expect the invariable presence of a pitch rise during the decay of every cone.

17.4. The Action of Piano Hammers

General principles were developed in chapter 8 to
guide our understanding of vibration recipes produced when strings are struck
ac various places by various kinds of hammers. In particular we found that the
duration of contact in a hammer blow exerts a considerable influence on the
number of characteristic modes that are excited. Modes having frequencies high
enough that one or more of their oscillations could cake place during the
contact time are, as a result, only weakly excited. On a piano the time of
contact is only partly influenced by the softness of the hammer felt; the
predominant influence arises from the way the string itself pushes back against
the hammer. Our thinking about this influence can conveniently be divided into
what we might call an elastic version and a wavelike version, the second
version being used to refine our conclusions from the first.

If someone were to force a piano
ham­mer slowly and progressively into the exaggerated position shown in the
upper part of figure 17.4, the tension of the deflected string would act on the
hammer, exerting a downward restoring force whose magnitude would grow as the
hammer is displaced farther and farther upward. It should be apparent from the diagram that the greater slant of
the shorter, left-hand segment of the scrim; means that this segment exerts the
major portion of the restoring force. For example, on a piano string whose
hammer strikes at a distance H equal to 1/9th of the string length L, the two
forces are in the ratio (L - H) / H =8 / 1, meaning that in this particular
case the restoring force of the short segment acting on the hammer will be
eight times as great as that of the long segment.

If one strikes a piano
key, the system of levers called the action accelerates the hammer to some
final speed and then releases it, allowing it to continue freely upward until
it strikes the string. When contact is made with the string, the hammer's
upward motion persists, but the string exerts an increasingly large downward
force on it as already described. If we temporarily set aside the force exerted
by the longer portion of the string, is it apparent that the hammer and the
shorter segment H of the string together constitute an elementary
spring-and-mass system. The natural frequency fH of this system is determined
by the string tension T, the length H, and the mass M of the hammer, as follows
(see sec. 6.1):

fH= (1/2š) xT/MH

The lower part of figure 17.4 shows the motion of the
hammer head (a) as it leaves its original rest position when the key is first pressed,
and then as it continues to accelerate under the influence of the player's
finger until the instant (b) when the action releases it. Following its release
by the action, the hammer swings freely upward toward the string and meets it
at (c), after which the string forces convert the motion into an up-and­ down
movement of oscillatory type (c), (d), (e). If the hammer were somehow to glue
itself now to the string, the oscillation would continue in the manner
indicated by the dotted curve and the letters (f), (g), (h), and (i). In fact,
the hammer comes loose from the string after about half a cycle of oscillation,
at the instant marked (e), and then falls back down until (j) when it is caught
and arrested by what is known as the check.

Clearly, if our calculation is
correct, the all-important time of contact T, between hammer and string is
about equal to one-half the time required for one oscillation of the hammer
bouncing on its "spring," which is the string length H. When we
modify the formula for ft, to take into account the three strings which act
together on any given hammer, we get:

Tc= (1/2)( 1/fH) =šÖMH/3T

Notice that, according to this formula, increasing
either the hammer mass M or the striking distance H will lengthen the time of
contact T, and thus reduce the number of higher partials excited in the tone,
as explained in chapter 8. The "elastic" version of the hammer recoil
analysis is now complete, and we must consider next how the wave behavior of
disturbances on the long segment of the string alters the conclusions we have
drawn thus far.

A hammer interacts with the longer
segment (L - H) of a piano string in a way that can easily be understood if we
begin by imagining the string to be extremely long, so that the hammer rebounds
from it before an echo returns from the far end. For instance, it would take
six seconds for an echo to return from the far end of a set of C4 piano strings
one kilometer long (about 0.6 mile). During the time the hammer is touching the
strings we have already noticed that it feels a springlike force exerted by the
short string segment H. Wave physics tells us that as the hammer launches waves
down the long segment of the strings, another force (in addition to the
springlike force) acts to make the hammer feel exactly as though it were
immersed in and plowing through an extremely viscous fluid. As a result, the
half-oscillation discussed earlier is damped (in the manner described in sec.
6.1). In this case the lost oscillatory energy is transmitted out along the strings
(for eventual return) instead of being frictionally dissipated. The viscous
dissipation coefficient D defined for a spring­mass system in section 6.1
proves to be exactly the aggregate wave impedance of the long segments of the
strings (see sec. 17.2)!

You may find it helpful to know
that the combined wave impedance of three C., strings is roughly equal to the
viscous coefficient D associated with two of your fingers moving broadside
through a howl of molasses left outdoors in January. Despite the numerically
large size of the vis­cous damping coefficient just described, calculation
shows that the wave-type damping on a piano hammer produces only a few percent
diminution in the amplitude of any one of
its oscillations, so that the formula of our original, simple estimate of the
hammer contact time T, does not yet need changing.

Having considered how the long
string segment feels to the hammer before an echo has time to return, we are
now ready to follow the progress of the half­ sinusoidal pulse impressed by the
hammer blow on the long side of the strings as it travels to the far end and
back. Fig­ure 17.5 indicates that a completed upward blow from the piano hammer
produces an upward pulse that travels to the bridge end and is then reflected
back toward the hammer. Because the bridge has a very large wave impedance
compared with char of the strings, this reflected pulse has very nearly the
same amplitude as the original, but it is inverted. As long as the hammer is
thrown clear in advance of the reflected wave (as is the case for notes below
about CS on a piano), the pulse runs back and forth over the whole length of
the strings, being reflected and re reflected ac the two ends. This particular
motion is exactly the one we described in chapter 8, using language that
derails the motion in terms of the vibration recipe belonging to a given hammer
blow at a particular point on the three strings (see secs. 8.2 and 8.3 and also
statement 6, sec. 11.9).

For the upper two octaves of the
piano scale, the inverted pulse returns before the hammer has left the strings,
and so adds its forces to those exerted by the short string segments H. As a
result, the hammer is thrown off the strings earlier than otherwise, thus
shortening the time of contact.

It is time now to go back and
refine our view of what is happening on the short length H of the strings
during the hammer blow. These do not really act exactly like the simple spring
we assumed originally. The disturbance on this is actually a peculiar train of
impulses rapidly echoing between the capo
d'astro bar and the hammer itself (ac C4 these impulses make about four
complete round trips during the time we calculated earlier for T,). When the
net effect of these rapid echoes is properly worked out, we find we must change
the half-sinusoidal hammer motion assumed earlier, which takes place between
(c) and (e) in figure 17.4. The hammer motion is now seen to have a new but
similar shape that looks as though is were made of roughly straight line
segments, each lasting the time is takes the impulses to make one round trip
between the fixed end and the hammer. The time of contact T, estimated earlier
remains fairly accurate, however, as do our earlier conclusions about the
effect on is produced by echoes coming back from the bridge.

The not-quite-sinusoidal
(segmented) hammer motion can be thought of as a combination of the original
sinusoid and an additional bouncing motion. This bouncing motion is of course caused
by string vibrations set up in the short string length H during the time of
hammer contact. These vibrations form a harmonic series whose frequencies are
L/H times as high as the corresponding modes of the complete, full-length
strings. During the course of the blow, then, the new high-frequency
oscillations of the hammer and of the short part of the string are given to the
complete strings in addition to the more familiar components of the vibration
recipe. It is somewhat shocking to
realize chat these extra components fill in the otherwise expected gaps in the
recipe produced when the hammer strikes at nodal points at various modes. For
example, in chapter 8 we learned that a simple, non segmented blow from a
mathematically idealized hammer 1/4th of the way from one end of a string would
eliminate modes 4, 8, 12, err., from the recipe. A real hammer blow restores
these missing components. We have here the explanation of the century-old
observation chat a piano-type hammer strikes in such a manner that no modes are
ever missing from the recipe of a piano tone. [5]

17.5. Scaling the Strings of a Piano

The piano that has been studied so painstakingly thus
far in the chapter would be of rather limited musical usefulness, for the
simple reason that is can do little more than play the note C4! The extension
of the basic design to the high and low limits of the scale is influenced by
constraints of a mechanical sort and also by the fact that our hearing changes
drastically as we go to these extremes. For example, the fundamental components
belonging to the top octave (from 2100 to 4200 Hz) span the most sensitive
range of our hearing, while the fundamentals of the lower notes (from 27.5 Hz)
are only weakly heard under ordinary playing conditions.

The formula given at the beginning
of section 16.5 for the vibration frequency of a flexible string suggests that
for every octave one goes up in pitch, the string length might be halved (if
the tension

and string size are kept fixed). We will see in a
moment why it proves better on a piano to reduce the lengths by a factor close
to 1/1.88 per octave, so chat if we start with our 62.5-cm C4 string, the Cs
string (four octaves higher) has a length close to 62.5/(1.88); = 5.00 cm,
instead of the 3.91-cm string length calculated on the basis of four successive
halvings. In a similar vein, experience has shown the advisability of reducing
the string diameter by a factor of about 0.946 per octave from the 1-mm
diameter at C4, making the cop string a little under 0.79 mm in diameter.
Strings proportioned thus have to be pulled to slightly lower tension ac the
top of the scale than at C4.

As we have already learned in
chapter 16, the inharmonicity of constant-tension strings proportioned in this
way rises 2.76-fold for every octave we go up. For example, at the top note
(Cs), mode 2 is more than 50 cents sharp compared with mode 1, instead of the
0.83-cent widening associated with C4. If many string partials for Cs were excited, the cone would be
quite harsh ("metallic," i.e., reminiscent of the vibrations of steel
bars), so the softness of the hammer felt must be carefully adjusted to give a
suitable contact time during the blow in order to produce a tone of acceptable
quality.

I will say little about the trend
relating the upper strings' wave impedance to chat of the soundboard beyond
remarking that at Ca the string impedance is only about 75 percent of the value
at C4, which reduces the transfer of vibration from string to soundboard by the
same factor. This reduction ac the upper end of the scale is desirable; not
only does the maker raise the soundboard impedance by using pro­gressively
shorter inactive string lengths (between bridge and hitch pin) to increase
their elastic contribution to the sound­board's own driving-point wave
impedance, he also tends to increase it even further by thickening the
soundboard at the treble end. Perhaps the increasing sensitivity of our ears
for higher-frequency sounds calls for a reduction in the actual amount of
vibration transmitted to them by the topmost strings. Another reason for
reducing the string-to-soundboard coupling is that it leads, as we have already
seen, to longer ringing of these strings. Recall that even so, the sound from
these strings decays so rapidly that dampers are not normally provided for
them.

The true challenge to the piano
maker's skill lies in the notes below C3. Even on a full concert grand with an
overall measure of nine feet, the bottommost strings must be made less than
half the lengths implied by the scaling rules used above Ca if the instrument
is to have dimensions less than those of a battleship. For example, the bottom
note (A) on a Baldwin concert grand I have examined has a string length close
to 203 cm, instead of 486 cm. The same note on the six-foot-long model L
Steinway grand in my living room has a somewhat shorter length-a little over
137 cm. On some small spinet pianos, the bottom string is a troublemaking 95
cm, only 20 percent of the "ideal" length.

How does one strive to meet the
requirements for acceptable (if not good) tone, sufficient power, and adequate
duration of sound in the lower strings? The need for acceptable tone implies
not only a tolerably low value for the string inharmonicity factor J, but also
a properly proportioned relationship among the hammer 's mass, breath, and
softness, the string tension, and the point at which the hammer strikes the
strings. To get sufficient power with an adequately long decay time one must in
addition arrange to get a correct ratio between the wave impedances of the
string and the bridge.

The formula given in section 16.5
for the natural frequencies of a flexible string suggests immediately that a
proportional increase in string thickness will automatically offset a reduction
in its length. For example, if we were to preserve our usual constant tension,
the bottom string of our concert grand would have a diameter that is 486/203
=2.39times the 1.22-mm diameter
of the fulllength string called for by our basic midscale design. Such a 3-mm
"string" would in fact be an impractically thick rod having nearly
half the diameter of the tuning pins! The inharmonicity of this rod would he so
large that it would emit a clanging sound when struck. The piano maker in
practice avoids a great deal of the inharmonicity problem by using a slender
steel string (to support the tension) which is wound with one or more layers of
copper wire, so as to raise the mass per unit length without adding much
stiffness. On a concert grand, carefully designed bass strings of this sort are
held under a tension that is about 50 percent larger than the mid ­scale value.
We find then (on the Baldwin concert grand, for example) that J calculated '
for the bottom string has a surprisingly low value, about equal to the
inharmonicity coefficient belonging to C3 in the main part of the scale. On the
smaller pianos, however, the problem remains serious, and what passes for good
tone cannot be obtained from a bottom string shorter than about 130 cm. On the
smallest spinets, J for the bottom string can he as much as ten times the
concert grand's value.

Of particular concern to the piano
maker is the problem of making a smooth transition from the full-length plain
wire strings to the sequence of shortened wound strings that function for the
lowest notes of the scale. Let us see how the problem is dealt with in the
Steinway mentioned earlier. On this instrument the lowest triple set of plain
wires is found at B2. The next note down the scale, B2♭, is provided with a pair of copper-wound wires having
a length of about 91 cm. I have used the wire sizes and playing frequencies of
these two sets of strings to calculate that the tension in the wound strings is
60 percent higher than in the plain ones, the latter being about 10 percent
slacker than normal because they are already 10 cm shorter than the basic
scaling rules would call for. The calculated string tension shows a rather
large jump, so I checked the correctness of this calculated change in tension
by comparing the pitches of sounds produced by plucking the non playing lengths
of these strings between bridge and hitch pin. The copper windings of the B2♭ string do not extend into this region, and
the core wire diameter is equal to the wire size for B2.

As one slowly plays down the
chromatic scale in the vicinity of the break between nonwound and wound
strings, one notices a slight but progressive deterioration of the tone below
F3, where the strings first begin to fall short relative to properly scaled
lengths of the sort used in the upper half of the instrument. The main
alteration in tone is due to a growing inharmonicity associated with both a
shortening of the string and the concomitant reduction of tension. The change
of tone one hears in going between wound and nonwound strings is relatively
small, the inharmonicity increase due to the greater stiffness of wound strings
being offset by the increase in their tension. The calculated inharmonicity
factors match within 5 percent across the break, which is less than the 9
percent change from note to note of the normal scaling!

The question of
suitable gradation of string and soundboard wave impedance across the break is
our next concern. Because the wave impedance depends on Ö [(inertia) x (elasticity)], it is apparent that in going
from the "properly scaled" F3 down to the slightly slack strings at
B2, we have a reduction of about 5 percent in the string impedance. Across the
break, from the three slack wires at B2 to the two wound and very tight strings
at B2♭, there is a 40 percent upward jump in wave impedance.
Let us resort once again to observation in order to find out how the maker has
dealt with these non­ uniformities.

On the Steinway in my home there
is a trend of progressive increase in the decay times measured for sounds from single strings belonging to the notes
running from F3 down to B2. The falling sequence of wave impedances for these
strings, all mounted in a row on the same bridge, leads us to expect a 5 percent
increase in decay time, roughly the observed amount. We must now compare the
decay time for a single wound string belonging to B2♭ with a plain string belonging to B2; these two
strings are found to vibrate for roughly the same lengths of time. On the other
hand, the fact that the B2♭ strings have a
40-percent the string lead to piano tones that, for notes from the bottom of
the scale tip to about C2, are made up of many (20 or 30) partials hav­ing
significant strength in the room. That is, these partials average out to have
pressure amplitudes that are more than 3 to 5 percent of the average amplitude
of the three or four lowest-frequency partials.

2. Between G2 and C5 the sound
pressure erciee typically contains partials of appreciable amplitude only up to
about 3000 Hz. This means that at G2, 25 or 30 partials are tonally
significant; at G3 the number is about halved; at C5 only 5 or 6 partials play
much of a role in the sound.

3. For notes above C5 (fundamental
frequency near 523 Hz) the number of significant partials decreases
progressively, until at C8 (near 4200 Hz) the fundamental is accompanied by
very little more than the second partial.

4. For notes below C2,
the first few partials of a tone may have roughly equal amplitudes; however, because
of the insensitivity of the ear below 60 Hz, the lowest-frequency partials of
these tones contribute very little to the perceived loudness.

5. Along with the components
having sharply defined frequencies (associated with the string vibrations), the
piano produces a very considerable amount of fairly diffuse sound made up of
closely spaced, even overlapping frequency components arising from (a) the
thumping blow of the hammer as transmitted to the frame by the short part of
the string and to the soundboard by the initial impulse carried to it by the
longer string segment, (b) the short-lived oscillations of all the damped
strings that are impulsively excited via the bridge motion at the beginning of
the tone, (c) the analogous, higher ­frequency sounds associated with the
"inactive" string lengths between bridge and soundboard, and (d) the
excitation of fairly long lived oscillations in the topmost strings of the
scale, which are not provided with dampers. Predominant among the sounds described
in (c) is the contribution from the strings whose long part has been struck by
the hammer. In many pianos the frequency of this contribution is harmonically
related to that of the played note.

6. Due in part to the momentary compression
and hardening of hammer felt in the course of a vigorous hammer blow, the
vibration recipes associated with fortissimo
playing show an augmentation of the strengths of the higher partials
relative to the lower ones. This means that the number of significant partials
in a given tone is increased when it is loudly played. The converse behavior is
observed at the limits of pianissimo playing.

7. After a key is struck, the
reverberative sound builds up in the soundboard, whence it radiates into the
room. Each component of the sound comes out in its own way, but when our ears
consider the sound in the aggregate, they are chiefly sensitive to the fact
that there is an initial burst of sound associated with the combined effects of
the struck strings and the various short-lived components listed in statement
5. At C4 this burst builds up in roughly 0.03 seconds. During the next few
tenths of a second the short-lived components disappear, leaving the string
sounds to decay in the manner described below. The time scale of these
developments is longer when low notes are sounded and shorter for high notes.
This is in part because as we go from one part of the scale to another, the
predominantly excited components have different frequencies and therefore different
"spreading times" across the soundboard.

8. The decay of any given partial
in a piano tone has a complicated nature (see secs. 17.2 and 17.3). Taking the
sound as heard overall, we find that 20 seconds is a good round number for the
time required for the aggregate sound pressure of C4 to fall to 1/1000th of the
original sound pressure. As we go up to about G7, the corresponding decay times
become progressively shorter, decreasing by a factor of about 0.66 for each
octave. Above G7 the decay times become very much shorter, falling to a value
of less than 0.5 seconds at Cg. Going
down the scale below C4, we find that the decay times become gradually longer,
growing by a factor of about 1.2 for each octave.

9. When a key is released, the
damper falls into place, but it is unable to kill off the string oscillations
immediately. Each partial decays at its own race, because of the varying
efficacy of damper action on each string mode. A characteristic part of the piano sound is the quickly damped tail
at the end of each tone. The fundamental component disappears in a few tenths
of a second, usually leaving the higher partials softly audible for several
seconds.

10. Both top and bottom surfaces
are active in coupling the soundboard vibrations to the air. When the cover is
lowered, there is a distinct change in piano tone, because the top surface of
the soundboard now communicates with the broad but not very high channel of air
between it and the lid. Signals produced by parts of the soundboard far back in
this channel take longer to come out into the room than do those which arise in
regions near the open edge of the lid. The impulsive parts of the piano tone
are influenced by the fact that in a narrow (almost two-dimensional) channel of
air, a given frequency component is transmitted in several "propagation
modes" each having its own (frequency - dependent) velocity. As a result
of the altered coupling to the room, the build-up processes described in
statement 7 are rearranged and spread over a longer period of time. Piano
makers recognize the existence of a lid angle giving "best" tone in a
concert hall. Curiously enough, the props installed on the piano to adjust the
lid angle do not always provide this "best" angle.

17.7. Examples, Experiments, and Questions

1. Find a freshly tuned piano and play the bottom note A0
along with its octave, A1. Get a fairly clear impression in your ears of the
joint sound of these two carefully adjusted notes. Next listen to sounds produced
by the two strings when you lightly touch the exact center of the A0 string
with a finger tip so that its odd ­numbered modes are quickly damped out. You
will be surprised at how many hitherto inaudible beats become easily apparent,
producing a very rough sound. When the string is damped in its exact center,
only those partials of A0 remain which directly
compare themselves with the partials of A1 (see the digression on sounds
with only even harmonics in sec. 5.5, and also experiment 2, sec. R.6). If a
piano tuner were present, he could retune A0 so as to make the beats less
noticeable in the altered circumstances, but then the normal A0-to-A1 octave
tuning would become unacceptable. See if you can find some acoustical reasons
why the presence of the odd-numbered partials in the normal tone of A0 leads
the tuner to provide a setting different from the elementary minimum-beat
conditions which we might naively expect. At the bottom of the keyboard, tuners
generally check not only the octave, but also double octaves, as they proceed.
They usually play their pairs of notes alternately as well as together. Why is
this an admirable practice?

2. The number of round
trips per second (shall we call it the echoing rate?) which a disturbance can
make on a string of length L is v/2L, where v is the wave velocity for
disturbances on this string. Replace the v in this echoing rate formula v/2L
with the formula for the wave velocity of a flexible string given in section
17.1. Verify that the resulting formula for the echoing rate is exactly the
same as the formula set forth at the beginning of section 16.5 for the mode-1
vibration frequency of a flexible string. To say there is a pulse echoing back
and forth on a string is simply another way of saying that a repetitive process
is going on, one which therefore can be described in terms of a set of
harmonically related sinusoidal components. These components are the ones we
have studied so much already from quite a different viewpoint. They are of
course associated with the various vibrational modes of the string. In the real
world of piano strings, the wire has stiffness, which means we cannot (strictly
speaking) calculate a simple echoing rate. This is because a given impulse does
not preserve its shape as it travels along the string, so that the disturbance
on the string is not of a strictly repeating nature. As a result its recipe
cannot be based on harmonically related frequencies. The reason for the
changing form of the traveling impulse is that the various sinusoidal
components making it up do not all travel at the same speed on a stiff string
(see the remark about wave speed on a soundboard in sec. 17.1). It is the stiff
string characteristic frequencies, therefore, which are the proper basis for
constructing a vibration recipe.

3. A number of practical
advantages are obtained from the practice of multiple stringing on pianos. Some
very fine pianos are built using four instead of three strings on each note in
the main part of the scale. You may find it interesting and worthwhile to
speculate about the acoustical consequences of adding strings in this way.
Assume first that no other changes
are made in the instrument, and consider such matters as hammer rebound time,
string impedances, tuning spread, etc., as they influence the overall tone. Why
would it be very difficult to arrange for the use of multiple strings at the
extreme bass?

4. Flat steel ribbons used instead
of sets of round wires would appear to allow the piano maker convenience in adjusting
the relations between inharmonicity, wave impedance, and the effect of string
tension on hammer rebound. He would have ribbon thickness, width, and length at
his disposal. The simple string frequency formula in this case becomes fn
=(n/2L) Ö [T/ (rwd)] , where t is the ribbon thickness and w is its width. See if
you can verify that the wave impedance becomes Ö Ttwd. The inharmonicity coefficient J uses the
thickness t instead of wire radius r (the width w being irrelevant), and the
numerical factor takes on a slightly larger magnitude. See if you can come up
with a list of arguments for and against the use of tape like strings in a
piano.

5. Many people find it possible to pick out fleeting parts of a
musical sound if their ears have once been told what to listen for.

The various impulsive sounds
coming at the beginning of a piano tone may be separated one from another by
your ears if you will produce each one by itself before listening for it in a
normally produced sound. For example, knock repeatedly at some point on the
bridge by tapping with the eraser end of a vertically held pencil, and become
familiar with the woody thump that results. Listen for this same thump when you
play a vigorous note on some key whose strings pass over the bridge at your
thumping point. Play only very short notes so that the sustained part of the
tone does not distract you. Notice that the pitch of the thump varies with the
position of the striking point along the keyboard. Now run your fingers lightly
and quickly across a range of playing strings. Do this near the bridge and
listen for the brief, harp like glissando that results. If you run upward in
pitch along the top octave of damped strings it may be possible to imagine that the sound is reminiscent of someone
whispering the word "whee !" After running your fingers back and
forth across these strings and sounding them at random, you will probably be
able to pick out their collective sounds when one of these triplets of strings
is struck a short, sharp blow with its hammer in the normal way. The ringing of
the damped strings is easily audible, and will be quite recognizable. Now see
if you can hear these same sounds when the piano key is held down after each
blow. Other aspects of the piano's initial tone can be tracked down with the
help of similar ear-training experiments. You will also find it worth­while to
listen to a number of notes, each one being played by itself and allowed to die
away. Damping one or two of the strings belonging to the individual note will
produce tonal changes which will repay your close attention.

6. In 1949, Franklin Miller, Jr.,
of Kenyon College proposed that adding one or more lumps of material to a piano
string might usefully reduce its inharmonicity and so improve the tone (see
sec. 9.6, part 1)[8]. The reduction of frequency produced by loading a wire is
one-half the shift associated with a similar loading on a plate (see sec. 9.4,
assertion 3). You may wish to experiment by wrapping a few centimeters of wire
solder tightly around a piano bass
string near one end or the other. Why will wrapping it at a point 1/ 10th of
the way along produce the desired result, a progressively greater alteration
(lowering) of the first five modes? From your knowledge of hammer dynamics you
may be able to decide whether it is better to add the load at the bridge end of
the string or at the hammer end. Even though the steady ­state motion (made up
of the modified string vibrations) can be made more nearly repetitive (i.e.,
more harmonic) by the use of an added mass, the initial echoing after the
hammer blow will include among other things an early echo due to partial
reflection from the added mass plus a later, modified pulse returning from the
string termination. How will this affect the early impulsive part of the piano
tone? Do you think that a normal bass string has its inharmonicity raised or
lowered by the fact char the copper windings do not extend all the way to the
two ends?

5. Hermann Helmholtz, On the Sensations of Tone, mans.
Alexander Ellis from 4th German ed. of 1877 with material added by translator
(reprint ed., New York: Dover, 1954). See Appendix V, "On the Vibrational
Forms of Pianoforte Strings," and Appendix XX, section N, part 2,
"Harmonics and Partials of a Pianoforte String Struck at One ­eighth of
Its Length." For references to more mod­ern work see A. B. Wood, A Textbook of Sound. 3rd rev. ed. (1955;
reprinted., London: Bell, 1960), pp. 100-101, and E. G. Richardson, ed., Technical Aspects of Sound, 3 vols.
(Amsterdam: Elsevier,352 1953, 1:464-68.

The reader
should be cautious in accepting the correctness of some of the work referred to
by Wood and Richardson. To make matters worse, the summary of it given by
Richardson is garbled at several points.

6. This calculation is based on equation 28c in the
paper by Harvey Fletcher, "Normal Vibration Frequencies of a Stiff Piano
String," J. Acoust. Soc. Am. 36
(196-I): 203-9. The effect of a
winding taper at the string ends is incorrectly given by Fletcher as the
explanation of certain inharmonicities that are in fact due to soundboard
resonances.

7. Some of the information presented here is taken
from the article by Daniel W. Martin (see note 4 in this chapter). Most of the
remainder is to be found in Harvey Fletcher, E. Donnell Blackham, and Richard
Stratton, "Quality of Piano Tones," /. Acoust. Soc. Am. 3- (1962): 749411. A less technical account of
this same work is given by E. Donnell Blackham, "The Physics of the
Piano," Scientific American. December
1965, pp. 88-99. The dependability of some of the conclusions drawn in these
two papers is reduced by the fact that one of the pianos was badly out of tune
at the time the experiments were done, thus deranging the excitation and decay
processes of the strings, as well as the frequencies of their partials.

The Clavichord and the
Harpsichord

The clavichord and the harpsichord developed earlier
than the piano. We will devote only limited attention to the first of these,
chiefly as a way to help us use our
knowledge of piano-hammer dynamics to understand some of the tonal attributes
of the harpsichord.

18.1. The Clavichord

On a clavichord each key forms a simple rocking lever
whose far end carries a wedge-shaped metal tangent
that rises up against the string. We might think of the tangent as being a
narrow and hard piano hammer, but unlike a piano hammer, the tangent is not
released by the action to fall away from the string. The sounding length of a
clavichord string is the part between tangent and bridge, while the short
portion of the string (marked H in the top of fig. 17.4) is provided with a
strip of damping felt to keep it more or less silent. We can use the lower part
of figure 17.4 to help ourselves visualize what happens when a clavichord key
is pressed. The tangent (carrying its own mass plus those of the key and the
player's finger) rises toward the string along the curve (a) (b) (c), exactly
as does the hammer on a piano. Once contact is made, the string and tangent remain
together and move tip and down in an oscillatory manner very similar to what is
indicated by the letters (d), (e), . . . (i). The chief difference between what
is sketched in the figure and the motion of a tangent is that the steady
pressure of the player's finger causes the whole bouncing oscillation to take
place above and below an average position somewhat above the original string
position (c). Also, this oscillation is fairly heavily damped (chiefly by the
player's finger). The formula in section 17.4 for the oscillating frequency
f" of a hammer bouncing on its string applies to the present situation as
well, and we can understand from it why the large mass M of
tangent-plus-key-plus-finger joins with the clavichord's low string tension T and
relatively long, felt­ damped string length H to give the tangent a very low
bouncing frequency. The period of oscillation (1/ffl) of this bouncing is
several tenths of a second instead of the few hundredths of a second that it is
on a piano.

The tangent's oscillation is
exceedingly slow in comparison to the string's musical vibrations. The initial
kink imposed on the string at the instant of contact can therefore be thought
of as echoing rapidly back and forth along a string whose "fixed" end
at the tangent is gradually moving upward and downward. According to wave
physics, the string modes that combine to give such a motion follow a recipe
that is almost identical with the recipe of a string plucked very close to one end by a narrow
plectrum. In other words, all the lower-numbered modes decrease with mode
number very nearly as 1/n (instead of 1/n2) up to that mode whose wiggliness
has a curvature matching the curve produced
by the stiffness of the string at the plucking point (see sec. 7.3, sec. 7.4,
part 6, and sec. 8.4).

The actual vibration recipe of a
clavi­chord string is not quite like that described in the preceding paragraph.
During the earliest instants after the tangent touches the string, the contact
force is not very large, so that each of the first few echoes returning from
the bridge actually makes the string jump off the tangent momentarily once per
cycle of the lowest string mode. These jumps are rap­idly damped out, however,
because at each jump a considerable amount of wave energy escapes past the
tangent to he eaten up by the felt damper on the string beyond. What we hear
then is a very brief "tzip" of sound at the beginning of each tone.
The components of this initial sound are of course in exact harmonic relationship to the echo repetition rate. The
sustained portion of the clavichord sound quite resembles the tone of a
harpsichord, though it is considerably softer.

18.2. The Harpsichord

The harpsichord has enjoyed a long period of
popularity that extends to the present day. Its development began well before
1600 and peaked during the first half of the eighteenth century. Even in the
early part of this period the art of harpsichord building and design was well
developed and sophisticated, and the harpsichord gave way to the piano only when
the latter instrument was improved to the point where it became competitive.
Fine harpsichords made as long ago as 1618 by the Ruckers family show much of
the subtlety of soundboard-ribs-and-bridge design and string scaling that we
find in today's pianos. Wire of brass, iron, and steel was available in
accurately graded sizes for use by early harpsichord builders. The predominant
British and German system of the eighteenth century gave 9.4 percent reductions
in going from one numbered size to the next, so that there were eight wire
sizes for each doubling of diameter, quite enough to take care of the necessary
changes for scaling harpsichord strings.'

Let us make a quick survey of the
relation between the note C9 on a particular Ruckers harpsichord and the same
note on a typical grand piano of today. On the old instrument, the string
length is slightly greater-70 cm instead of 62.5-as comports with a lower
overall tuning based on an A4 setting near 410 Hz. The steel strings for this
note have a diameter of 0.32 mm instead of the piano's 1 mm, and the
harpsichord's string tension is about 11 percent of that used on a piano. This
reduction in string tension is almost entirely attributable to the use of
thinner strings rather than to limitations of strength in the materials.
Typically, for each octave one goes up from C4, the strings are 50 percent as
long as the corresponding ones in the lower octave, in contrast to 53 percent
on the piano. The diameter of a harpsichord string will be 73 percent of the
measure of its mate an octave lower, while on a piano the higher note has
string diameters that are 94 percent of those for the note an octave below it.
As we go down from mid-scale, the bass strings of a harp­sichord grow according
to the rule described for the upper scale. The sound­board on this harpsichord
has a thickness varying between 2.5 and 3 mm, in contrast to a thickness near
10 mm that is typical of a piano.'

The mounting of light, low-tension
strings on a thinner soundboard gives a string-to-soundboard wave impedance
ratio for the harpsichord that is higher by a factor of 1.3 than the ratio
between a single string and the piano soundboard. This might lead us to expect
the decay time on a harpsichord to be about 20/1.3 = 15.4 seconds (see statement 8, sec. 17.5; why is it
correct here to use the single-string rather than the triple-string wave
impedance for the piano?). I find by informal trial on a similarly proportioned
modern harpsichord that the apparent persistence of the overall tone is in fact
much less than this, the time being on the order of half a dozen seconds.
However, the above prediction based on wave impedance ratios is oversimplified,
since it does not take into account the damping of string vibrations by viscous
friction in the surrounding air. This damping has only a small role to play in
the behavior of a piano string, but it cannot be ignored on the harpsichord.

In 1856 the distinguished British
physicist Sir George Stokes worked out the theory of such air damping of string
vibrations by viscous friction, and showed among other things that the
vibrations of small-diameter wires are more quickly damped than are chose of
large wires (halving the diameter halves the decay time). He further showed
that the high­ frequency modes die away more quickly than do the lower ones
(doubling the frequency reduces the decay time by about 70 percent). If we
confine our attention to the lowest-frequency component (mode 1) of sounds from
the two instruments, it is possible to reconcile their decay times reasonably
well by including the effects of air friction.' Numbered statement 4, below,
implies the resolution of any remaining discrepancy in the perceived decay
times.

Digression on Archimedes and Mersenne.

The inverse relationship
between vibration frequency and both string length and diameter was recognized
long ago by Archimedes (287-212 B.C.), at least in the tense that halving
either dimension would raise the pitch by an octave. The fondness of Greek
intellectuals and their successors in Europe for simple ratios at a means for
expressing the perfection of nature obscured for many years the fact that the
vibrations of bars and those of water in a cup do not follow such
relationships, and (in particular) also obscured the square-root relationship
between string tension and vibrating frequency. 1t it perhaps significant that
the Frenchman Marin Mersenne (1588-1648), who is credited today with
scientifically clarifying the nature of string vibrations, !iced at a time when
craftsmen were already very expert in making use of musical strings. Mersenne
was quite aware of the influence of stiffness on the effective lengths of
strings. We should realize, however, that the class of ideas implied by our
term wave impedance did not become well systematized until the
latter half of the nineteenth century, following the laying of the Atlantic
telegraph table. [4]

We are now in a position to
describe the tonal nature of the harpsichord sound.

1. When a harpsichord key is
depressed, the plectrum is in contact with, the string for a short time before
the string slips off of it to vibrate freely. During this short contact time,
small-amplitude but audible clavichord-like vibrations arc set up on the
portion of string between plectrum and bridge, and also in the part between
plectrum and fixed string end. In particular, the sound begins with a brief but
complex buzz as the echoing impulses on both sides of the string cause it to
tap against the plectrum. The sound recipe also contains harmonic components
belonging to the characteristic vibrations of the short and long portions of
the string acting independently. These are not generally in tune with the note
eventually to be produced, the exact frequencies depending on the position of
the plucking point along the string.

2. Once the jack has pulled the
string aside and released it, ordinary plucked-string vibrations of thesore discussed in sections 7.3, 8.1, and 8.4
are set up on the whole string. Furthermore, the string shapes and velocities
chat are present on the two sides of the jack before the string slips clear now
become free to travel up and down the length of the entire string. The presence
of these additional vibrations means chat. as in the case of the piano tone,
the complete recipe has in it modes of vibration that have nodes at the
plucking point.

3. When the player releases a key,
the plectrum brushes past the string slightly before the damper comes down into
action. During this interval of time an extra bit of sound arises from the
momentary tapping (buzzing) of the string as the plectrum slips past it.
Because this tapping takes place between the string and a relatively hard.
narrow object, a great many of the string modes are excited to appreciable
amplitude. This is particularly true because, in contrast to the effect of a
single. metallic tangent blow, we have here repeated blows, all exactly in step
with the natural vibrations of the string. The duration of the tapping
excitation is somewhat longer than that of its clavichord like predecessor
during initial plucking. The brief chirp that one generally hears at the end of
a harpsichord tone is compounded out of the main tone plus the components added
on the plectrum's return trip, these being permitted to decay over a period of
1/4 to 1/2 a second after the relatively narrow damper comes into action.

4. Besides the expected brief
ringing after the damper touches the string, there is one more aspect of the
damped sound of a harp­sichord string that helps to establish the musical
personality of the instrument. Since the damper is firm and narrow, the segment
of string between the fixed end and the damper vibrates briefly at its own
natural frequencies. In general the pitch of this short sound is not in any
musical relation to the main tone. One finds, however, that certain strings of
a harpsichord scale have their dampers located close to a node for one of their
higher partials (typically the 5th, 6th, or 7th). For these strings, then, our
ears arc provided with a more lingering, harmonically related reminder of the
main tone, which may last for a second or two.

5. The tone color and (to a slight
extent) the loudness are both altered when a key is struck more or less hard.
That is in part due to changes in the amount and duration of the clavichordlike
fraction of the tone. The remaining contribution comes from changes in the
relative amounts of soundboard and damped-string sound that are produced in
comparison with the relatively fixed amplitudes of the main string sounds (see sec.
17.6, statement 5).

6. The sound pressure recipe for a
harpsichord note contains a much larger number of important partials than does
the tone of a piano. The effect of frictional damping by the air on the slender
strings of a harpsichord causes the high-frequency components of its tone to
die away very much more quickly than do the lower partials, so that the
perceived duration of the tone as a whole is very short. During the decay, the
tone color changes because the vibration recipe rapidly loses its higher
partials, exactly the reverse of the way in which piano tones are heard to
survive longest via their higher partials.

7. Due to the
slenderness of harpsichord strings, the inharmonicity of the partials of a
harpsichord tone is generally very much less (e.g., l/14th as large at C4) than
that found on a piano. The musical effect of the greater harmonicity is not
particularly apparent, however, because partials 2, 4, 6, . . . of the
harpsichord tone have, very crudely speaking, the same frequency shifts at C4
due to inharmonicity as do partials 1, 2, 3, . . . of the piano tone. The
harpsichord's larger string impedance relative to that of its bridge also
increases the random inharmonicity due to soundboard resonances. The
harpsichord tone thus collects by means of its large number of important
partials an aggretate inharmonicity that does not differ much from the
inharmonicity associated with the fewer partials in a piano tone. Tuning
discrepancies are perhaps a little harder to detect in the more diffuse but
shorter-lived sound of a harpsichord.

18.3. Examples, Experiments, and Questions

1. On large harpsichords one finds stops that give the
player a choice of varying tone colors. One of these stops arranges for the
strings to be plucked a considerable distance from their fixed ends. Another
arrangement presses a small block of felt against each string very near to the
fixed end, so as to damp its vibrations lightly. See how many musical
implications you can draw from the acoustical changes produced by these stops.
Consider in particular that the influence of the added felt block at the end of
the string increases progressively as we go to the higher modes.

2. Most harpsichords have at least a pair of strings
for each note of the main scale, and the player has the option of plucking one
or both of these. For both mechanical and tonal reasons, the distance between
the fixed string end and the plucking point is different fits the two strings. If both strings of a pair could be
plucked exactly together, the acoustical consequences would be very similar to
those associated with multiple stringings on a piano. In practice the strings
are not released precisely together, which at the very least eliminates the
rapid initial decay. Think about the auditory consequences of having; strings
excited a few hundredths to one-tenth of a second apart. Consider next what
goes on if only one string of a closely tuned pair is excited directly by the
player, the ordinary damper being lifted for both. Each vibrational mode of the
plucked string their drives its originally silent counterpart into transient
motion to the sort described in chapter 10. As. the driven oscillations of the
second or "sympathetic" string build Lip, they in turn start driving
the soundboard, and so produce a certain share of the audible sound. See if you
can figure out why the vibrations of the plucked string may die out very
rapidly at first, and yet leave its with an actual swelling of audible sound.
What sort of tonal effect would you expect from the fact that the sympathetic
string, receives an initial impulsive excitation when the first kink arrives at
the bridge after the plectrum slips free of the other string?

3. Harpsichords are
usually provided with a set of so-called "four-foot" strings in
addition to the normal "eight-foot" ones that provide the basic
scale. The four-foot strings are tuned to sound an octave above their nominal
note names. Thus the C4 key of a harpsichord key­board can pluck strings tuned
to 261.6 Hz and also one tuned to 523.2 Hz. On the Rockers harpsichord
described earlier, the string lengths of the four-foot strings are
approximately half those of their eight-foot brothers. The wire sizes are not
quite the same, however: at C6 the wires are alike, at C4 the higher-pitched
string is about 10 percent thinner than the lower, while at C2 (the bottom
note), the four-foot string is about 20 percent smaller. The thinner,
high-pitched set of strings thus runs at a reduced tension, so that at C6 the
wave impedance is 90 per­cent of the eight-foot value; at C4 and at C2 the
figures are close to 80 and 70 percent. See what you can predict about the
loudness and sustaining power of the four-foot strings, taking account of the
fact that we hear better at high frequencies and also considering the fact that
the short strings run over their own slender bridge, after which they are
anchored with large downbearing directly to the soundboard in the region
between the two bridges. The four-foot strings are generally plucked a little
closer to their centers than is customary for the full-size strings.

4. Some concerts of baroque music
are played at today's pitch, based on A-440, while at other times the choice
favors a reference frequency near 415 Hz, which is a semitone lower.
Harpsichords are sometimes built so that either pitch can be selected by
mechanical transposition, the keyboard being slid sideways to operate on
different plectra and strings. On other instruments it becomes necessary to
retune the strings themselves to shift from high pitch to low pitch or vice
versa.

Consider what happens to a satisfactory high-pitch
instrument when the scring tension is slackened about 12 percent to bring it to
the lower tuning. What will happen to the decay time of the tones and to their
loudnesses? (Be careful here-there are several aspects to the physics and also
to the perception process.) How will the stiffness and soundboard­ resonance
contributions to the inharmonicity be changed? What musical consequences will
these have? The initial thump from the soundboard and frame will have an
audibly different relationship to the main sound. A serious builder of
harpsichords might find similar cogications useful in suggesting ways to guide
his proportioning of string gauges and lengths, soundboard thickness, bridge
and rib dimensions, downbearing angles, etc., when he adapts a successful
design intended for one tuning to the construction of an instrument tuned to
the other pitch.

5. Unlike the piano, the bottom
surface of a harpsichord case is closed by a large board traversed by several
stiffening ribs mounted on its inner surface from one side plank of the case to
the other. Soundboard vibrations communicated to the somewhat compartmentalized
air cavity within the case are radiated into the room via the long, narrow
opening left at the keyboard end of the soundboard. The overall balance of
sound from different parts of a harpsichord scale (both loudness and tone
color) can be influenced by the acoustical relationship of the air cavity modes
to those of the soundboard (see sec. 17.6, part 10, concerning the lid on a
piano). The effect is particularly noticeable at the bass end of the scale.
Refer back to section 9.5 and see how much of the discussion there of the
interaction of kettledrum cavity and drumhead can be adapted to the present
situation. Why could the cutting of a hole in the case bottom, or of an
elaborately carved "rose" in the soundboard, be expected to produce
tonal changes?

6. One occasionally meets notes on
a harpsichord that "beat with themselves" even when only a single
string is permitted to vibrate. The simplest way in which this phenomenon can
come about is the following. At the bridge, the stiffness of the anchorage
appears considerably greater to a string that vibrates from side to side in a
horizontal plane (parallel to the soundboard) than it does to a string
vibrating more normally in a vertical plane. Because of this, any given mode of
oscillation will have a slightly higher frequency when excited in a horizontal
plane than in a vertical (see sec. 16.5). Furthermore, we find that the plane
of oscillation for such a string will slowly rotate, at a rate equal to the
frequency difference between the two versions of the mode. As a result, an
initial vertically oriented oscillation produced by normal plucking will slowly
rotate a quarter revolution into a horizontal oscillation (which cannot drive
the bridge) before continuing another quarter revolution, at which time it will
again become a vertical oscillation, etc. Verify char the sound waxes and
wanes, as a result of this rotation, at twice the frequency we normally would
associate with ordinary beats between the modes. Why is this whole phenomenon
most likely to manifest itself for string modes whose frequencies lie close to
resonances of the soundboard? Can you figure out why even a slight kink put
into a wire during installation can give rise to a similar kind of bearing
sound?

4. A fascinating account of the
historical development of these ideas is to be found in Sigalia Dostrovsky,
"Early Vibration Theory: Physics and Music in the Seventeenth
Century," Archive for History of
Exact Science, in press.